This book, first published in 1971, is about what we sometimes call “special functions”. It emphasizes special functions of particular interest in physics. Many of these arise naturally as solutions to ordinary differential equations. Yet there are a bewildering number of “standard” special functions and most of them have more than one definition. It can be quite a challenge for an author to get the reader past the feeling that the subject is nothing but a chaotic collection of formulas. This book attempts to do that through careful selection and organization. It is by no means a comprehensive study of special functions. Hochstadt instead chose his topics according to his estimation of their value in mathematical physics, and to some extent to follow his own interests.

Most of the subjects treated here arose from questions studied in the eighteenth or nineteenth centuries by the likes of Gauss, Euler, Bessel and Legendre, and nearly all of them were motivated by physical problems. Now, when we tend to fussier about the distinctions between pure and applied mathematics, we’d say that this book is split between topics of interest primarily in applied mathematics and topics of purely mathematical interest.

The major topics are orthogonal polynomials (two chapters), hypergeometric functions (also two chapters), the Gamma, Legendre and Bessel functions, and spherical harmonics. The final chapter, a bit of an outlier, focuses on Hill’s equation. This was first used to describe the stability of the moon’s orbit and later applied to the motion of an electron in a crystal. When this book was written, Hill’s equation was not well known in the literature, and this chapter was partly intended to fill that gap.

Hochstadt does not assume any prior knowledge of special functions, but he does expect the reader to be very comfortable with real and complex analysis at or above the advanced undergraduate level. Although the presentation is directed in part at physicists, engineers and other applied scientists, the treatment occasionally pushes the envelope with some of the discussions (for example, Riemann surfaces and Schwarz-Christoffel transformations). The writing is fluid and arguments are generally easy to follow. However, there is very little attempt to put the discussions of special functions in context or explain where and why they might be used. So, if readers don’t know why they should care about hypergeometric functions, they’re not going to learn it here. This is nonetheless a valuable reference, a great place to browse or send students to learn about orthogonal polynomials, Legendre polynomials, Bessel functions and the like.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.